Theorem 3 ([1]}, [2]}) 
The images in \(\mathbb {R}^n\)  for \(n=0,1,2, \ldots \)  under projection maps \(\mathbb {R}^{n+k}\Rightarrow \mathbb {R}^n\)  of sets with the form \(\lbrace (x,y)\in \mathbb {R}^{n+k}:P(x,y,e^x,e^y)=0\rbrace \)  is definable, where \(P\)  is a real polynomial in \( 2(n+k)\)  variables with \(x:=(x^1, \ldots ,x^n)\) , \(y:=(y^1,\ldots ,y^k)\) , \(e^x:=(e^{x^1}, \ldots ,e^{x^n})\)  and \(e^y:=(e^{y^1},\ldots ,e^{y^k})\) .

